# semilatent: Estimate Graphical Models with Latent Variables and... In latentgraph: Graphical Models with Latent Variables

## Description

Estimate graphical models with latent variables and replicates using the method in Tan et al. (2016).

## Usage

 `1` ```semilatent(data, n, R, p, lambda, distribution = "Gaussian", rule = "AND") ```

## Arguments

 `data` data set. Can be a matrix, list, array, or data frame. If the data set is a matrix, it should have nR rows and p columns. This matrix is formed by stacking n matrices, and each matrix has R rows and p columns. If the data set is a data frame, the dimention and structure are the same as the matrix. If the data set is an array, its dimention is (R, p, n). If the data set is a list, it should have n elements and each element is a matrix with R rows and p columns. `n` the number of observations. `R` the number of replicates for each observation. `p` the number of observed variables. `lambda` tuning parameter that encourages estimated graph to be sparse. `distribution` For a data set with Gaussian distribution, use "Gaussian"; For a data set with Ising distribution, use "Ising". Default is "Gaussian". `rule` rules to combine matrices that encode the conditional dependence relationships between sets of two observed variables. Options are "AND" and "OR". Default is "AND".

## Details

The semilatent method has two assumptions. The first one states that the latent variables are constant across replicates. Assumption 2 states that given the latent variables, the replicates are mutually independent. With these two assumptions, the method seeks to solve the following problem for 1 ≤ j ≤ p.

\min_{β_{j,O / j}} \{l_j (β_{j,O / j}) + λ\|β_{j,O / j}\|_1 \},

where l_j (β_{j,O / j}) is a nuisance-free loss function, β_{j,O / j} is a parameter that represents the conditional dependence relationships between jth observed variable and the other observed variables, and λ is a tuning parameter. This method aims at modeling semiparametric exponential family graphical model with latent variables and replicates.

## Value

 `omega` a matrix that encodes the conditional dependence relationships between sets of two observed variables `theta` the adjacency matrix with 0 and 1 encoding conditional independence and dependence between sets of two observed variables, respectively `penalty` the penalty value

## References

Tan, K. M., Ning, Y., Witten, D. M. & Liu, H. (2016), ‘Replicates in high dimensions, with applications to latent variable graphical models’, Biometrika 103(4), 761–777.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```#semilatent Gaussian with "AND" rule n <- 50 R <- 20 p <- 30 seed <- 1 l <- 2 s <- 2 data <- generate_Gaussian(n, R, p, l, s, sparsityA = 0.95, sparsityobserved = 0.9, sparsitylatent = 0.2, lwb = 0.3, upb = 0.3, seed)\$X result <- semilatent(data, n, R, p, lambda = 0.1,distribution = "Gaussian", rule = "AND") ```

latentgraph documentation built on Dec. 15, 2020, 5:23 p.m.